# Pricing of forwards contracts

Of the courses I am taking in college this semester, two are Financial Mathematics and Derivatives. In each course, we learn different formulas to calculate the forward price of a forward contract. Obviously, the two formulas must equal each other, but I am not sure what the link is i.e. how to get from one formula to the other.

In my Derivatives course, I was taught that $$F_0 = S_0 e^{(r - d)T},$$ where

$$F_0$$ is the forward price,

$$S_0$$ is the spot price,

$$r$$ is the risk-free rate,

$$d$$ is the average yield per annum and

$$T$$ is the length of the contract (in years).

In my Financial Mathematics course, it is also given that $$F_0 = (S_0 - PV_I) e^{\delta T},$$ where

$$F_0$$ is the forward price,

$$S_0$$ is the spot price,

$$PV_I$$ is the present value of the fixed income payment(s) due during the term of the contract,

$$\delta$$ is the force of interest and

$$T$$ is the length of the contract (in years).

How do I reconcile the two formulas? In other words, how can I prove that $$F_0 = S_0 e^{(r - d)T} = (S_0 - PV_I) e^{\delta T}?$$

Any intuitive explanations will be greatly appreciated! :)

## 1 Answer

Decompose the first formula as $$F_0=(S_0 - S_0(1-e^{-dT}))e^{rT}$$ then let $$PV_{I} = S_0(1-e^{-dT})$$ which represents the present value of dividends (dividend rate = $$d$$) paid on the security during the life of the contract, and you obtain the second formula.

• Hold on... If I expand $S_0 - S_0(1 - e^{-dT})$, doesn't the $S_0$s cancel out? How can that be right? Might there be a typo somewhere? – user52981 Mar 5 at 8:24
• cancels out to $S_0 e^{-dT}$ and you recover the original formula – Antoine Conze Mar 5 at 8:28
• Oh yes. Sorry brain is not functioning... Also, does this mean that the force of interest is equivalent to the risk-free rate, since I can see that $\delta = r$? – user52981 Mar 5 at 9:02